Analysis of Variance (ANOVA) is a statistical
method used to test differences between two or more means. It may seem odd that
the technique is called "Analysis of Variance" rather
than "Analysis of Means."
As you will see, the name is appropriate because inferences about
means are made by analyzing variance.

ANOVA is used to test general rather than specific
differences among means. This can be seen best by example. In the
case study "Smiles
and Leniency," the effect of different types
of smiles on the leniency shown to a person was investigated.
Four different types of smiles (neutral, false,
felt, miserable) were investigated. The chapter
"All Pairwise Comparisons
among Means" showed how to test
differences among means. The results from the Tukey HSD test
are shown in Table 1.

Table 1. Six Pairwise Comparisons.

Comparison

Mi-Mj

Q

p

False - Felt

0.46

1.65

0.649

False - Miserable

0.46

1.65

0.649

False - Neutral

1.25

4.48

0.010

Felt - Miserable

0.00

0.00

1.000

Felt - Neutral

0.79

2.83

0.193

Miserable - Neutral

0.79

2.83

0.193

Notice that the only significant difference is
between the False and Neutral conditions.

ANOVA tests the non-specific null hypothesis that
all four population means are equal. That is,

μfalse = μfelt = μmiserable = μneutral.

This non-specific null hypothesis is sometimes
called the omnibus null hypothesis.
When the omnibus null hypothesis is rejected, the conclusion is
that at least one population mean is different from at least one
other mean. However, since the ANOVA does not reveal which means
are different from which, it offers less specific
information than the Tukey HSD test. The Tukey HSD is therefore
preferable to ANOVA in this situation. Some textbooks introduce
the Tukey test only as a follow-up to an ANOVA.
However, there is no logical or statistical reason why you should
not use the Tukey test even if you do not compute an ANOVA.

You might be wondering why you should learn about
ANOVA when the Tukey test is better. One reason is that there
are complex types of analyses that can be done with ANOVA and
not with the Tukey test. A second is that ANOVA is by far the
most commonly-used technique for comparing means, and it is important
to understand ANOVA in order to understand research reports.